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Let $W$ be a standard one dimensional Brownian motion, and $\mathcal F_t$ its natural filtration. Consider the SDE

$$dX_t = \mu_X (t, \omega) \, dt + \sigma_X (t, \omega) \, dW_t$$

$$dY_t = \mu_Y (t, \omega) \, dt + \sigma_Y (t, \omega) \, dW_t$$

$$X_0 = x_0, Y_0 = y_0 \text{ a.s.}$$

where $\mu_X, \mu_Y, \sigma_X, \sigma_Y \geq 0$ are progressively measurable with respect to $\mathcal F_t$, and $x_0, y_0$ are constants.

Assume the existence of a solution to the above two equations up to a determinstic time $T$.

Question: Suppose $\sigma_X \neq \sigma_Y$ on a subset of $\Omega \times [0, T]$ of positive measure. Then is it true that

$$\mathbb P(Y_T > X_T) > 0?$$

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Don't think so. Take both to satisfy something like $dZ = \sigma |Z| dW$, start one from -1, and one from 1. regardless of the exact parameters, one stays positive and one stays negative.

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  • $\begingroup$ Ugh damn you’re right.. $\endgroup$
    – Nate River
    Jul 21, 2022 at 12:13

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